On the Standard Model of Particle Physics

Prelude

Since our discussion of the Standard Model will center around its Lagrangian, it seems prudent to remember some
of the reasons we are so interested in a Lagrangian formulation in the first place. From a Lagrangian function

L(φi(x), ∂μ φi(x))

and the associated action

S = ∫ L d4x

we can obtain (see, i.e., Itzykson and Zuber, Quantum Field Theory):

the Euler-Lagrange equations of motion, by requiring the action to remain stationary under variations in the fields

δ S / δ φi(x) = 0 →

∂L / ∂φi(x) - ∂μ (∂L / ∂(∂μ φi(x))) = 0;

the canonical energy-momentum tensor, from requiring the action to be stationary under xμ → xμ + εμ:

Tμν = ∂L / ∂(∂μ φi(x)) (∂ν φi(x)) - L

which is conserved (∂μ Tμν = 0);

conserved currents from Noether's Theorem, which requires the action to be stationary under field variations with respect to an internal
symmetry group (defined by generators Ta):

δS (φi(x) → εa(x) Ta φi(x)) = 0

→ jμa(x) = ∂L(φ + δφ) / ∂(∂μ εa(x));

and the accompanying conserved charges:

Qa = ∫ d3x j0a(x);

and perhaps most importantly (at least for perturbation theory), the Feynman rules for the primitive interaction vertices:
each term in the Lagrangian specifies either propagators (kinetic terms) or possible fundamental interactions between the
various field components.

It is for this last reason that we will be particularly interested in examining the types of terms included in:

It is probably not possible to overestimate the importance of the covariant derivative and how gauge invariance limits the types of terms
which are possible. Just as in General Relativity the covariant derivative allows us to define a consistent idea of the tangent space,
in the standard model it allows us to consistently define isospin and color space, and hypercharge. And it is responsible for:

the existence of bosonic derivative couplings (labeled "momentum factor" in the table below), and the lack of such couplings for fermions; and

the fact that all 2W, 3W and 4W vertices are isospin neutral, and that all 3- and 4-gluon vertices are color neutral
(note that no 2-gluon vertices occur, because the Higgs are SU(3) scalars).